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Order Statistic Tree
In computer science, an order statistic tree is a variant of the binary search tree (or more generally, a B-tree) that supports two additional operations beyond insertion, lookup and deletion: * Select(''i'') – find the ''ith smallest element stored in the tree * Rank(''x'') – find the rank of element ''x'' in the tree, i.e. its index in the sorted list of elements of the tree Both operations can be performed in worst case time when a self-balancing tree is used as the base data structure. Augmented search tree implementation To turn a regular search tree into an order statistic tree, the nodes of the tree need to store one additional value, which is the size of the subtree rooted at that node (i.e., the number of nodes below it). All operations that modify the tree must adjust this information to preserve the invariant that size = size .html"_;"title="eft[x">eft[x_+_size[right[x_+_1 where_size[nil.html" ;"title="">eft[x_+_size[right[x.html" ;"title=".html" ;"title="eft ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical disciplines (including the design and implementation of Computer architecture, hardware and Computer programming, software). Computer science is generally considered an area of research, academic research and distinct from computer programming. Algorithms and data structures are central to computer science. The theory of computation concerns abstract models of computation and general classes of computational problem, problems that can be solved using them. The fields of cryptography and computer security involve studying the means for secure communication and for preventing Vulnerability (computing), security vulnerabilities. Computer graphics (computer science), Computer graphics and computational geometry address the generation of images. Progr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Search Tree
In computer science, a binary search tree (BST), also called an ordered or sorted binary tree, is a rooted binary tree data structure with the key of each internal node being greater than all the keys in the respective node's left subtree and less than the ones in its right subtree. The time complexity of operations on the binary search tree is directly proportional to the height of the tree. Binary search trees allow binary search for fast lookup, addition, and removal of data items. Since the nodes in a BST are laid out so that each comparison skips about half of the remaining tree, the lookup performance is proportional to that of binary logarithm. BSTs were devised in the 1960s for the problem of efficient storage of labeled data and are attributed to Conway Berners-Lee and David Wheeler. The performance of a binary search tree is dependent on the order of insertion of the nodes into the tree since arbitrary insertions may lead to degeneracy; several variations of the b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
B-tree
In computer science, a B-tree is a self-balancing tree data structure that maintains sorted data and allows searches, sequential access, insertions, and deletions in logarithmic time. The B-tree generalizes the binary search tree, allowing for nodes with more than two children. Unlike other self-balancing binary search trees, the B-tree is well suited for storage systems that read and write relatively large blocks of data, such as databases and file systems. Origin B-trees were invented by Rudolf Bayer and Edward M. McCreight while working at Boeing Research Labs, for the purpose of efficiently managing index pages for large random-access files. The basic assumption was that indices would be so voluminous that only small chunks of the tree could fit in main memory. Bayer and McCreight's paper, ''Organization and maintenance of large ordered indices'', was first circulated in July 1970 and later published in '' Acta Informatica''. Bayer and McCreight never explained what, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Selection Algorithm
In computer science, a selection algorithm is an algorithm for finding the ''k''th smallest number in a list or array; such a number is called the ''k''th '' order statistic''. This includes the cases of finding the minimum, maximum, and median elements. There are O(''n'')-time (worst-case linear time) selection algorithms, and sublinear performance is possible for structured data; in the extreme, O(1) for an array of sorted data. Selection is a subproblem of more complex problems like the nearest neighbor and shortest path problems. Many selection algorithms are derived by generalizing a sorting algorithm, and conversely some sorting algorithms can be derived as repeated application of selection. The simplest case of a selection algorithm is finding the minimum (or maximum) element by iterating through the list, keeping track of the running minimum – the minimum so far – (or maximum) and can be seen as related to the selection sort. Conversely, the hardest case of a se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Best, Worst And Average Case
In computer science, best, worst, and average cases of a given algorithm express what the resource usage is ''at least'', ''at most'' and ''on average'', respectively. Usually the resource being considered is running time, i.e. time complexity, but could also be memory or some other resource. Best case is the function which performs the minimum number of steps on input data of n elements. Worst case is the function which performs the maximum number of steps on input data of size n. Average case is the function which performs an average number of steps on input data of n elements. In real-time computing, the worst-case execution time is often of particular concern since it is important to know how much time might be needed ''in the worst case'' to guarantee that the algorithm will always finish on time. Average performance and worst-case performance are the most used in algorithm analysis. Less widely found is best-case performance, but it does have uses: for example, where the b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-balancing Binary Search Tree
In computer science, a self-balancing binary search tree (BST) is any node-based binary search tree that automatically keeps its height (maximal number of levels below the root) small in the face of arbitrary item insertions and deletions.Donald Knuth. ''The Art of Computer Programming'', Volume 3: ''Sorting and Searching'', Second Edition. Addison-Wesley, 1998. . Section 6.2.3: Balanced Trees, pp.458–481. These operations when designed for a self-balancing binary search tree, contain precautionary measures against boundlessly increasing tree height, so that these abstract data structures receive the attribute "self-balancing". For height-balanced binary trees, the height is defined to be logarithmic \mathcal O(\log n) in the number n of items. This is the case for many binary search trees, such as AVL trees and red–black trees. Splay trees and treaps are self-balancing but not height-balanced, as their height is not guaranteed to be logarithmic in the number of items. S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loop Invariant
In computer science, a loop invariant is a property of a program loop that is true before (and after) each iteration. It is a logical assertion, sometimes checked within the code by an assertion call. Knowing its invariant(s) is essential in understanding the effect of a loop. In formal program verification, particularly the Floyd-Hoare approach, loop invariants are expressed by formal predicate logic and used to prove properties of loops and by extension algorithms that employ loops (usually correctness properties). The loop invariants will be true on entry into a loop and following each iteration, so that on exit from the loop both the loop invariants and the loop termination condition can be guaranteed. From a programming methodology viewpoint, the loop invariant can be viewed as a more abstract specification of the loop, which characterizes the deeper purpose of the loop beyond the details of this implementation. A survey article covers fundamental algorithms from many a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
AVL Tree
In computer science, an AVL tree (named after inventors Adelson-Velsky and Landis) is a self-balancing binary search tree. It was the first such data structure to be invented. In an AVL tree, the heights of the two child subtrees of any node differ by at most one; if at any time they differ by more than one, rebalancing is done to restore this property. Lookup, insertion, and deletion all take time in both the average and worst cases, where n is the number of nodes in the tree prior to the operation. Insertions and deletions may require the tree to be rebalanced by one or more tree rotations. The AVL tree is named after its two Soviet inventors, Georgy Adelson-Velsky and Evgenii Landis, who published it in their 1962 paper "An algorithm for the organization of information". AVL trees are often compared with red–black trees because both support the same set of operations and take \text(\log n) time for the basic operations. For lookup-intensive applications, AVL trees ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Red–black Tree
In computer science, a red–black tree is a kind of self-balancing binary search tree. Each node stores an extra bit representing "color" ("red" or "black"), used to ensure that the tree remains balanced during insertions and deletions. When the tree is modified, the new tree is rearranged and "repainted" to restore the coloring properties that constrain how unbalanced the tree can become in the worst case. The properties are designed such that this rearranging and recoloring can be performed efficiently. The re-balancing is not perfect, but guarantees searching in O(\log n) time, where n is the number of entries. The insert and delete operations, along with the tree rearrangement and recoloring, are also performed in O(\log n) time. Tracking the color of each node requires only one bit of information per node because there are only two colors. The tree does not contain any other data specific to it being a red–black tree, so its memory footprint is almost identical to that o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weight-balanced Tree
In computer science, weight-balanced binary trees (WBTs) are a type of self-balancing binary search trees that can be used to implement dynamic sets, dictionaries (maps) and sequences. These trees were introduced by Nievergelt and Reingold in the 1970s as trees of bounded balance, or BB �trees. Their more common name is due to Knuth. Like other self-balancing trees, WBTs store bookkeeping information pertaining to balance in their nodes and perform rotations to restore balance when it is disturbed by insertion or deletion operations. Specifically, each node stores the size of the subtree rooted at the node, and the sizes of left and right subtrees are kept within some factor of each other. Unlike the balance information in AVL trees (using information about the height of subtrees) and red–black trees (which store a fictional "color" bit), the bookkeeping information in a WBT is an actually useful property for applications: the number of elements in a tree is equal to the si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ICALP
ICALP, the International Colloquium on Automata, Languages, and Programming is an academic conference organized annually by the European Association for Theoretical Computer Science and held in different locations around Europe. Like most theoretical computer science conferences its contributions are strongly peer-reviewed. The articles have appeared in proceedings published by Springer in their Lecture Notes in Computer Science, but beginning in 2016 they are instead published by the Leibniz International Proceedings in Informatics. The ICALP conference series was established by Maurice Nivat, who organized the first ICALP in Paris, France in 1972. The second ICALP was held in 1974, and since 1976 ICALP has been an annual event, nowadays usually taking place in July. Since 1999, the conference was thematically split into two tracks on "Algorithms, Complexity and Games" (Track A) and "Automata, Logic, Semantics, and Theory of Programming" (Track B), corresponding to the (at lea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Python (programming Language)
Python is a high-level, general-purpose programming language. Its design philosophy emphasizes code readability with the use of significant indentation. Python is dynamically-typed and garbage-collected. It supports multiple programming paradigms, including structured (particularly procedural), object-oriented and functional programming. It is often described as a "batteries included" language due to its comprehensive standard library. Guido van Rossum began working on Python in the late 1980s as a successor to the ABC programming language and first released it in 1991 as Python 0.9.0. Python 2.0 was released in 2000 and introduced new features such as list comprehensions, cycle-detecting garbage collection, reference counting, and Unicode support. Python 3.0, released in 2008, was a major revision that is not completely backward-compatible with earlier versions. Python 2 was discontinued with version 2.7.18 in 2020. Python consistently r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
